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Properties

Label 2-605-55.13-c1-0-12
Degree $2$
Conductor $605$
Sign $0.425 - 0.905i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.35 − 0.690i)2-s + (−0.787 − 0.124i)3-s + (0.182 − 0.251i)4-s + (0.834 + 2.07i)5-s + (−1.15 + 0.374i)6-s + (0.0299 + 0.189i)7-s + (−0.401 + 2.53i)8-s + (−2.24 − 0.730i)9-s + (2.56 + 2.23i)10-s + (−0.175 + 0.175i)12-s + (1.40 + 2.75i)13-s + (0.171 + 0.235i)14-s + (−0.398 − 1.73i)15-s + (1.39 + 4.30i)16-s + (−1.68 + 3.30i)17-s + (−3.54 + 0.562i)18-s + ⋯
L(s)  = 1  + (0.957 − 0.487i)2-s + (−0.454 − 0.0720i)3-s + (0.0912 − 0.125i)4-s + (0.373 + 0.927i)5-s + (−0.470 + 0.152i)6-s + (0.0113 + 0.0714i)7-s + (−0.142 + 0.896i)8-s + (−0.749 − 0.243i)9-s + (0.810 + 0.706i)10-s + (−0.0505 + 0.0505i)12-s + (0.389 + 0.764i)13-s + (0.0457 + 0.0629i)14-s + (−0.102 − 0.448i)15-s + (0.349 + 1.07i)16-s + (−0.408 + 0.800i)17-s + (−0.836 + 0.132i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.425 - 0.905i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.425 - 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48010 + 0.939800i\)
\(L(\frac12)\) \(\approx\) \(1.48010 + 0.939800i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.834 - 2.07i)T \)
11 \( 1 \)
good2 \( 1 + (-1.35 + 0.690i)T + (1.17 - 1.61i)T^{2} \)
3 \( 1 + (0.787 + 0.124i)T + (2.85 + 0.927i)T^{2} \)
7 \( 1 + (-0.0299 - 0.189i)T + (-6.65 + 2.16i)T^{2} \)
13 \( 1 + (-1.40 - 2.75i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (1.68 - 3.30i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (0.601 - 0.437i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.14 + 1.14i)T + 23iT^{2} \)
29 \( 1 + (-7.72 - 5.61i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.108 + 0.333i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.33 + 0.845i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (3.93 + 5.41i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (3.72 - 3.72i)T - 43iT^{2} \)
47 \( 1 + (-1.93 + 12.2i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-8.04 + 4.09i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (5.65 - 7.78i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.60 - 1.82i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (-4.13 + 4.13i)T - 67iT^{2} \)
71 \( 1 + (3.54 + 10.9i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.37 - 0.375i)T + (69.4 - 22.5i)T^{2} \)
79 \( 1 + (0.207 - 0.637i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-14.8 - 7.57i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 - 7.92iT - 89T^{2} \)
97 \( 1 + (0.626 + 1.22i)T + (-57.0 + 78.4i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.90634004349128336045802545553, −10.44024279781719913643206420376, −9.020050083000947836281025646351, −8.293238485198964627082517569181, −6.83175779924895901168174140361, −6.14743861822682527954664179127, −5.28727040661145901469471823316, −4.09792796190947200048090186977, −3.14678717402223990540990341235, −2.08174856488473034281491347481, 0.76425635019775829842704265941, 2.82753401176092802783892731095, 4.33081064744441893738578346587, 4.99676930789755751881228063561, 5.82302584807203116390546577186, 6.39649080830487844250666326669, 7.77731945038406451208370138515, 8.710660444431478363626415411561, 9.649752144144245577130017750590, 10.49153199485204966080716049573

Graph of the $Z$-function along the critical line